Certain properties of generalized and Higher–order q-Hermite polynomials: monomiality and applications to their zero distributions

In the present paper, we demonstrate 3-variable 2-parameter $q$-Hermite polynomials via generating functions along with their series definitions, $q$-derivatives, operational identities, then we deduce some properties for 2-variable 1-parameter $q$-Hermite polynomials. Also, we present the same mentioned features for multi-index $q$-Hermite polynomials and their associated formalism. Moreover, we utilize the techniques of quasi-monomial extension to explain and implement $q$-multiplicative and $q$-derivative operators for $q$-Hermite polynomials in three variables and multi-index $q$-Hermite polynomials. Finally, we present applications that can be derived using these polynomials, where the graphs of the zero functions and the meshes are displayed.